The ink cartridge needed for a HP 1040 is the HP 20 cartridge. Remove it from its packaging and remove the protective tape from the copper printhead. Be careful not to touch the printhead.
Press the Access Cartridge button on the control panel of the fax machine. This will position the cartridge carrier to the install position.
Pull the control panel toward the front of the machine to open it. Then open the cartridge compartment cover by lifting it toward the back of the machine.
Remove the old cartridge by pulling it toward the front of the machine. Place the new cartridge in the carrier and push the top of the cartridge until it clicks into place.
Close the cartridge compartment cover and control panel. Press the Start/Enter button. Press 1 for a new cartridge or 2 for a used one.